Limit of the Right-Most Position of a Modified Branching Random Walk

Abstract: In this talk, we consider a modification of the usual Branching Random Walk (BRW), where at the last step we give certain displacements which may be different from the increments. Under very minimal assumption on the underlying point process we show that the maximum displacement converges to a limit after only an appropriate centering. We further show that the centering term is $c_1 n + c_2 \log n$ and give explicit formula for the constants $c_1$ and $c_2$. If the underlying point process is i.i.d. displacements then we show that $c_1$ is exactly same and $c_2$ is $1/3$-of the corresponding constants of the usual BRW. Our proofs are based on a novel method of coupling with a more well studied process known as the smoothing transformation.

[This is a joint work with Partha Pratim Ghosh]